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Full text of "Analytical Mechanics"

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FIELDS OF FORCE AND NEWTONIAN POTENTIAL 217 4. Find the potential and the field intensity due to a straight circular rod at a point on its axis. 5. Show that problems 2 and 3 are special cases of problem 4. 6. Find the magnetic potential and the field intensity due to a cylindrical magnet at a point on its axis; suppose the magnetism to be distributed at the ends only. 7. Find the potential and the field intensity due to two spherical charges at a point equidistant from centers of the two charges. 8. Find the potential and the field intensity due to a right cone at a point on its axis. 9. A uniform solid sphere is cut in two by a diametral plane. Show that the gravitational force between the two parts will be — 7-1, where 16 a2 m is the mass of the sphere, a the radius, and 7 the gravitational constant. 10. Show that if any two points on the surface of the earth were joined by a straight and smooth tunnel a particle would traverse it in about 42.& minutes. 11. Two spheres of masses m and m' attract each other with a forcey F ~ 7 —;r > w-^ere 7 is a constant and r is the distance between the centers.. Taking the configuration when the spheres are in contact to be that of zero potential energy, find their potential energy when the centers are separated by a distance D. The radii of the spheres are a and &. 12. In the preceding problem suppose the spheres to repel each other with the same law of force and take the configuration when the spheres are separated by an infinite distance to be that of zero potential energy. 13. Find the potential due to a small magnet at a point whose distance is large compared with the length of the magnet, 14. In the preceding problem find the components of the intensity of the field along and at right angles to the line joining the point to the magnet. Also find the total intensity and its direction.